Liouville's theorem for a fractional elliptic system

Pengyan Wang; Pengcheng Niu

doi:10.3934/dcds.2019067

Article Contents

2019, Volume 39, Issue 3: 1545-1558. Doi: 10.3934/dcds.2019067

This issue Previous Article Symmetry for an integral system with general nonlinearity Next Article Symmetry properties in systems of fractional Laplacian equations

Liouville's theorem for a fractional elliptic system

Pengyan Wang and
Pengcheng Niu^,

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China

* Corresponding author: Pengcheng Niu
* Corresponding author: Pengcheng Niu

Received: December 31, 2017

Revised: April 30, 2018

Published: December 2018

The authors are supported by the National Natural Science Foundation of China (No.11771354), China Postdoctoral Science Foundation (No.2017M613193)and Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University

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Abstract

In this paper, we investigate the following fractional elliptic system

$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$

where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.

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Mathematics Subject Classification: Primary: 35A01, 35B53, 35J61; Secondary: 35B09.

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